Core Concepts

This note provides a complete classification of the "quality" of non-compactness for embeddings between Besov spaces, characterizing when they are finitely strictly singular and strictly singular, which could indicate the existence of "most optimal" target spaces.

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Chuah, C. Y., Lang, J., & Yao, L. (2024). Note about non-compact embeddings between Besov spaces. arXiv preprint arXiv:2410.10731.

This note investigates the "quality" of non-compactness for embeddings between Besov spaces, aiming to classify when these embeddings are finitely strictly singular and strictly singular.

Key Insights Distilled From

by Chian Yeong ... at **arxiv.org** 10-15-2024

Deeper Inquiries

It's certainly possible to explore extending the techniques used in the note to other function spaces. Here's a breakdown of the key techniques and potential challenges:
Key Techniques Used:
Wavelet Decomposition: This is a central tool for Besov spaces, allowing for their characterization in terms of sequence spaces. This simplifies the analysis of embeddings.
Reduction to Sequence Spaces: By using wavelet isomorphisms, the authors translate the problem of studying embeddings between Besov spaces to studying embeddings between sequence spaces like ℓq(ℓp) spaces.
Analysis of Bernstein Numbers: Bernstein numbers are used as a measure of "quality" of non-compactness. The authors analyze these numbers for the sequence space embeddings.
Diagonal Theorem: This theorem helps to lift results about finite strict singularity from individual embeddings between sequence spaces to embeddings between spaces of sequences.
Extending to Other Function Spaces:
Spaces with Similar Decompositions: The techniques could be applicable to function spaces that admit decompositions analogous to wavelet decompositions. Examples include:
Triebel-Lizorkin spaces: These spaces are closely related to Besov spaces and share similar wavelet characterizations.
Modulation spaces: These spaces use a different type of decomposition based on the short-time Fourier transform, but the principle is similar.
Challenges for Other Spaces:
Lack of Suitable Decompositions: Spaces without convenient decompositions would require different approaches.
Complexity of Embeddings: The analysis of Bernstein numbers might become significantly more complex for spaces with less tractable structures.
Finding Analogous "Optimal" Embeddings: The concept of "most optimal" embeddings, linked to non-strict singularity in the note, might need to be redefined or adapted for different function space settings.
In summary, while the techniques hold promise for spaces with similar decomposition properties, extending them to a broader class of function spaces would require careful consideration of the specific structures and challenges posed by those spaces.

Yes, there are alternative ways to characterize "most optimal" target spaces that don't solely depend on strict singularity. Here are a few possibilities:
Interpolation Theory:
Optimal Interpolation Spaces: Given an embedding X ↪ Y, one can seek the "smallest" or "most optimal" space Z such that X ↪ Z ↪ Y, where Z is an interpolation space between X and Y. This often involves finding the interpolation space with the smallest possible parameters.
K-Functionals and Interpolation Inequalities: The optimality of Z can be related to the behavior of the K-functional associated with the interpolation couple (X, Y). Sharp interpolation inequalities can also provide insights into the optimality of embeddings.
Rearrangement-Invariant Spaces:
Optimal Target within a Class: As mentioned in the note, the embedding into Lp∗,p(Ω) is optimal within the class of rearrangement-invariant spaces. One could explore optimality within other specific classes of function spaces relevant to the problem at hand.
Rearrangement Inequalities: Sharp rearrangement inequalities can be used to establish the optimality of embeddings within classes of rearrangement-invariant spaces.
Entropy Numbers and Other Approximation Quantities:
Minimal Decay of Entropy Numbers: Instead of strict singularity, one could focus on the decay rate of entropy numbers. An embedding with the slowest possible decay of entropy numbers (within a given context) could be considered "most optimal."
Other s-Numbers: Similar considerations could be applied to other s-numbers, such as approximation numbers or Gelfand numbers, to characterize optimality.
Geometric Properties of Target Spaces:
Minimizing Distortion: The "most optimal" target space could be the one that minimizes the distortion of the image of the embedding. This could involve studying properties like Banach-Mazur distances or other measures of geometric similarity between spaces.
The choice of the most appropriate characterization would depend on the specific function spaces involved and the desired properties of the "most optimal" target space in the given application.

Understanding the "quality" of non-compactness, particularly in the context of strict singularity, has significant practical implications in various applications involving Besov spaces:
1. Image Processing:
Image Compression: Non-compact embeddings are closely related to the ability to approximate functions (or images) in one space by elements of another space. Finitely strictly singular embeddings suggest the possibility of achieving good compression rates with relatively few coefficients.
Denoising and Regularization: In image denoising and regularization problems, Besov spaces are often used as regularization terms. The "quality" of non-compactness can influence the effectiveness of these regularization methods and the smoothness properties of the reconstructed images.
2. Partial Differential Equations (PDEs):
Existence and Regularity of Solutions: The compactness of embeddings between function spaces is crucial in proving the existence and regularity of solutions to PDEs. Understanding the precise nature of non-compactness can provide insights into the types of solutions that can be expected.
Numerical Analysis of PDEs: In numerical methods for PDEs, such as finite element methods, the "quality" of non-compactness can affect the convergence rates of the numerical schemes. Finer distinctions beyond simple compactness can lead to more accurate error estimates and better algorithm design.
3. Other Applications:
Approximation Theory: The study of non-compact embeddings is fundamental in approximation theory, where the goal is to approximate functions in one space by elements of another. Strict singularity and related concepts provide tools for analyzing the efficiency of such approximations.
Statistical Learning Theory: Besov spaces are used in statistical learning theory to characterize the complexity of function classes. The "quality" of non-compactness can influence the generalization properties of learning algorithms and the rates of convergence.
In essence, understanding the "quality" of non-compactness allows for:
More Refined Analysis: It goes beyond simply knowing whether an embedding is compact or not, providing a deeper understanding of the relationship between the spaces involved.
Improved Algorithm Design: It can guide the development of more efficient algorithms for image processing, numerical analysis, and other applications.
Sharper Theoretical Results: It leads to more precise theoretical results, such as improved error estimates and convergence rates.

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